Distributive double p-algebras whose congruence lattices are chains
A distributive double p-algebra \((A,\wedge,\vee,^*,^+,0,1)\) is a structure of type \((2,2,1,1,0,0),\) where \((A,\wedge,\vee,0,1)\) is a bounded distributive lattice and \({}^*(\vee)\) denotes pseudocomplementation (dual pseudocomplementation), i.e. \(y=x^*\) if and only if \(z\wedge x=0\) implies \(z\leq y\), and \(x^+\) is defined dually. The paper under review investigates the question, when \(Con A,\) the lattice of congruences of A, is a chain; earlier results in this direction have been obtained by \textit{R. Beazer} [Algebra Univers. 10, 225-231 (1980; Zbl 0431.06012)] and \textit{B. A. Davey} [ibid. 8, 73-88 (1978; Zbl 0381.06019)]. In particular, the author gives a new necessary condition for \(Con A\) to be a chain of cardinality 3, and shows that the condition is not sufficient. Also, several algebras A are constructed whose lattice of congruences is a five element chain; all previously known examples had smaller cardinality. Throughout the paper, Priestley's duality [\textit{H. A. Priestley}, Q. J. Math., Oxf. II. Ser. 26, 215-228 (1975; Zbl 0323.06013)] is applied, exhibiting the strength and beauty of this method.
scientific article; zbMATH DE number 3851170